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Vector Algebra

Question
CBSEENMA12036139

If straight u with rightwards arrow on top comma straight v with rightwards arrow on top comma straight w with rightwards arrow on top are non–coplanar vectors and p, q are real numbers, then the equality open square brackets 3 straight u with rightwards arrow on top comma straight p straight v with rightwards arrow on top comma straight p straight w with rightwards arrow on top close square brackets minus open square brackets straight p straight v with rightwards arrow on top comma straight w with rightwards arrow on top comma space straight q straight u with rightwards arrow on top close square brackets space minus space left square bracket 2 straight w with rightwards arrow on top comma straight q straight v with rightwards arrow on top comma straight q straight u with rightwards arrow on top right square bracket space equals space 0 holds for

  • exactly one value of (p, q)

  • exactly two values of (p, q)

  • more than two but not all values of (p, q)

  • all values of (p, q )

Solution

A.

exactly one value of (p, q)

3 straight p squared space left square bracket space straight u with rightwards arrow on top straight v with rightwards arrow on top straight w with rightwards arrow on top right square bracket minus pq left square bracket space straight u with rightwards arrow on top straight v with rightwards arrow on top straight w with rightwards arrow on top right square bracket space plus space 2 straight q squared left square bracket space straight u with rightwards arrow on top straight v with rightwards arrow on top straight w with rightwards arrow on top right square bracket space equals 0
rightwards double arrow space 3 straight p squared space minus space pq space plus 2 straight q squared space equals space 0
straight p space equals space fraction numerator straight q plus-or-minus space square root of 1 minus 24 end root straight q over denominator 2 space straight x space 3 end fraction
straight p space equals space fraction numerator straight q space plus-or-minus space square root of 23 space iq over denominator 2 space straight x 3 end fraction space rightwards double arrow space straight p space equals space 0 comma space straight q space equals space 0